Norm inequalities related to operator monotone functions

Let A,B be positive semidefinite matrices and any unitarily invariant norm on the space of matrices. We show for any non-negative operator monotone function f(t) on , and for non-negative increasing function g(t) on with g(0) = 0 and , whose inverse function is operator monotone. Received: 1 February 1999     

A note on the Hodge theory for functionals with linear growth

We study the Hodge decomposition of L ¹-(and measure-) differential forms over a compact manifold without boundary, giving positive results and counterexamples. The theory is then applied to the relaxation and minimization, in cohomology classes, of convex functionals with linear growth. This corresponds to a non-linear version of the Hodge theory, in the spirit of L. M. Sibner and R. J. Sibner [SS]. Received: 19 November 1997 / Revised version: 18 May 1998     

A method to exclude branch points of minimal surfaces

The central problem of this paper is to exclude boundary branch points of minimal surfaces. The method consists in showing that the third derivative of the Dirichlet energy is negative along well-chosen paths in admissible Jacobi field directions, if a “Schüffler condition” is satisfied. Received July 21, 1997 / Accepted October 3, 1997     

A characterization of graphs which can be approximated in area by smooth graphs

For vector valued maps, convergence in W ^1,1 and of all minors of the Jacobian matrix in L ¹ is equivalent to convergence weakly in the sense of currents and in area for graphs. We show that maps defined on domains of dimension n≥ 3 can be approximated strongly in this sense by smooth maps if and only if the same property holds for the restriction to a.e. 2-dimensional plane intersecting the domain. Received April 29, 1999 / final version received July 21, 2000?Published online September 25, 2000     

A comparison of the eigenvalues of the Dirac and Laplace operators on a two-dimensional torus

We compare the eigenvalues of the Dirac and Laplace operator on a two-dimensional torus with respect to the trivial spin structure. In particular, we compute their variation up to order 4 upon deformation of the flat metric, study the corresponding Hamiltonian and discuss several families of examples. Received: 10 December 1998     

Uniform partial regularity of quasi minimizers for the perimeter

Quasi minimizers for the perimeter are measurable subsets G of such that for all variations of G with and for a given increasing function such that . We prove here that, given , G a reduced quasi minimizer, and , there are , with , and , homeomorphic to a closed ball with radius t in , such that for some absolute constant . The constant above depends only on n, and . If moreover for some , we prove that we can find such a ball such that is a dimensional graph of class . This will be obtained proving that a quasi minimizer is equivalent to some set which satisfies the condition B. This condition gives some kind of uniform control on the flatness of the boundary and then criterions proven by Ambrosio-Paolini and Tamanini can be applied to get the required regularity properties. Received: July 12, 1999 / Accepted: October 1, 1999     

Variational sequences in mechanics

Let be a fibered manifold over a base manifold . A differential 1-form , defined on the -jet prolongation of , is said to be contact, if it vanishes along the -jet prolongation of every section of . The notion of contactness is naturally extended to -forms with . The contact forms define a subsequence of the De Rham sequence on . The corresponding quotient sequence is known as the rth order variational sequence. In this paper, the case of 1-dimensional base is considered. A simple proof is given of the fact that the rth order variational sequence is an acyclic resolution of the constant sheaf. Then the 1st order variational sequence is studied in detail. The quotient sheaves, as well as the quotient mappings, are determined explicitly, and their relationship to the standard concepts of the 1st order calculus of variations is discussed. The following is shown: a) the lagrangians in the 1st order variational sequence (classes of 1-forms) coincide with 2nd order lagrangians, affine in the second derivative variables, b) the concept of the Euler-Lagrange form is extended to 2-forms which are not necessarily variational, c) the concept of the Helmholtz-Sonin form is introduced as the class of an arbitrary 3-form, d) the well-known fundamental notions such as the Euler-Lagrange, and Helmholtz-Sonin mappings are represented by two arrows at the beginning of the variational sequence; this relates the global structure of the Euler-Lagrange mapping to the cohomology of , e) all the remaining classes of -forms with , as well as the quotient mappings, are determined explicitly, f) a locally variational form is defined as a generalization of a symplectic form; locally variational forms, associated to a fixed Euler-Lagrange form, are characterized, and g) distributions associated with a locally variational form are described, and their relation to the Euler-Lagrange equations is studied. These results illustrate differences between finite order variational sequences and variational bicomplexes, based on infinite jet constructions. Received February 18, 1996 / In revised form December 1996 / Accepted December 2, 1996     

Regularity for minimal boundaries in R n ¶with mean curvature in L n

Let be a minimal set with mean curvature in L ⁿ that is a minimum of the functional , where is open and . We prove that if then can be parametrized over the (n−1)-dimensional disk with a C ^α mapping with C ^α inverse. Received: 11 July 1997 / Revised version: 24 February 1998     

A new proof of regularity for¶two-shaded image segmentations

In this paper, we consider minimizing the Mumford-Shah functional over two-valued functions in the plane, which is equivalent to minimizing over characteristic functions. Existence of minimizers is straightforward and we show that any minimizing set is essentially open, has a boundary with finitely many connected components, and each component is C ¹. The relatively quick proof does not rely on quasi-minimal surface or varifold theory, but on uniform estimates on the regularity of the boundary. Received: 4 February 1997 / Revised version: 4 February 1998     

Homologically versus homotopically area minimizing surfaces in 3-manifolds

We construct a Riemannian metric on the 3-torus such that no closed surface minimizing area in its homology class is incompressible, i.e., each such surface is of genus greater than one. In particular, for such a Riemannian metric, the homotopically area minimizing 2-tori constructed in [5] do not minimize area in their homology classes. The example is easily generalized to arbitrary 3-manifolds. The constructed Riemannian metric can be chosen to be conformally equivalent to any arbitrary given one. Received September 4, 1998 / Accepted October 23, 1998     

Uniqueness of least area surfaces in the 3-torus

Given any -periodic metric g on and a plane through the origin, Bangert [4] shows that there exists a properly embedded surface homeomorphic to which is homotopically area-minimizing w.r.t. g, lies in a strip of bounded width around P, and does not have self-intersections when projected to the 3-torus . For the set of such surfaces, we show the following uniqueness theorems: If P is irrational, i.e., is not spanned by vectors in , the action of on by translations has a unique minimal set. If P is totally irrational, i.e., , then the surfaces in are pairwise disjoint. Received: 8 July 1999 / In final form: 14 February 2000 / Published online: 25 June 2001     

Computation of high order derivatives in optimal shape design

Summary. In shape optimization problems, each computation of the cost function by the finite element method leads to an expensive analysis. The use of the second order derivative can help to reduce the number of analyses. Fujii ([4], [10]) was the first to study this problem. J. Simon [19] gave the second order derivative for the Navier-Stokes problem, and the authors describe in [8], [11], a method which gives an intrinsic expression of the first and second order derivatives on the boundary of the involved domain. In this paper we study higher order derivatives. But one can ask the following questions: -- are they expensive to calculate? -- are they complicated to use? -- are they imprecise? -- are they useless? \medskip\noindent At first sight, the answer seems to be positive, but classical results of V. Strassen [20] and J. Morgenstern [13] tell us that the higher order derivatives are not expensive to calculate, and can be computed automatically. The purpose of this paper is to give an answer to the third question by proving that the higher order derivatives of a function can be computed with the same precision as the function itself. We prove also that the derivatives so computed are equal to the derivatives of the discrete problem (see Diagram 1). We call the discrete problem the finite dimensional problem processed by the computer. This result allows the use of automatic differentiation ([5], [6]), which works only on discrete problems. Furthermore, the computations of Taylor's expansions which are proposed at the end of this paper, could be a partial answer to the last question. Received January 27, 1993/Revised version received July 20, 1993     

Bifurcating critical points of bending energy under constraints related to the shape of red blood cells

Considered is a variational problem for the bending energy of closed surfaces under the prescribed area and surrounding volume. Minimizers of this problem are interpreted as surfaces modeling the shape of red blood cells. We give a rigorous proof of the existence of a one-parameter family of critical points bifurcating from the sphere and study their stability/instability. In particular, for a few branches of critical points, we compute the exact values of the index and the nullity of critical points. Received: 8 September 2001 / Accepted: 25 October 2001 / Published online: 29 April 2002 Partly supported by Grant-in-Aid for Exploratory Research (Nos.09874026, 11874033) and for Scientific Research (No.12640200), Ministry of Education, Science, Sports, and Culture, Japan; and also by Sumitomo Foundation Dedicated to Professor Takaaki Nishida on his sixtieth birthday     

Pôles de∫ A f s • pour une singularité presque isolée

Let be a germ of real analytic function (n≥ 1). We suppose that the complexified germ has an almost isolated singularity at 0 for an eigenvalue of the monodromy . Denote by A a linear combination of the connected components of . The purpose of this paper is to give a necessary and sufficient condition such that the distribution admits a sequence of poles in . Received: 18 July 1996 / Revised version: 13 May 1998     

Residual bounds of approximate solutions of the discrete-time algebraic Riccati equation

Let be a Hermitian matrix which approximates the unique Hermitian positive semi-definite solution to the discrete-time algebraic Riccati equation (DARE) where , is Hermitian positive definite, , the pair is stabilizable, and the pair is detectable. Assume that is nonsingular, and is stable. Let , and let be the residual of the DARE with respect to . Define the linear operator by The main result of this paper is: If where denotes any unitarily invariant norm, and then Received June 7, 1995 / Revised version received February 28, 1996     

Analysis of a part design procedure

Summary. In this paper we analyze and illustrate a new "ab initio" part design procedure, in which, given a cost function which reflects performance, materials, and manufacturing considerations, the topology and the geometry of the part are automatically produced. The analysis is based on demonstration of, first, the compactness of the metric space over which the cost function is defined, and, second, lower semi-continuity of the cost function. Examples include beams and elastic supports. Received November 15, 1993